Mathematical modeling

Monthly delivery to wholesaler 2Monthly delivery to wholesaler 3Monthly delivery to wholesaler 4Monthly delivery to wholesaler 5= 2700 units= 9000 units= 4500 units= 3600 unitsAssume x ij = number of units shipped from plant i to wholesaler jc ij = cost of shipping a unit from plant i to wholesaler jp i= production cost of a unit at plant iLet the variables be as follows:The variables are to determine the number of units to be shipped from eachproduction plant (the three plants) to each wholesaler (the five wholesalers); x ijThe objective is to determine the number of units produced and shipped from plant ito wholesaler j so that the cost of producing and shipping are minimized, costMin cost = (p 1 +c 11 )x 11 + (p 1 +c 12 )x 12 + (p 1 +c 13 )x 13 + (p 1 +c 14 )x 14 + (p 1 +c 15 )x 15 +(p 2 +c 21 ) x 21 +…………………or min cost = ∑∑( p i + c ij )x ij , i=1, 2, 3 and j=1, 2, 3, 4, 5The constraints are:∑ x i1 ≥ 2700, i=1, 2, 3 (amount to wholesaler 1)∑ x i2 ≥ 2700, i=1, 2, 3 (amount to wholesaler 2)∑ x i3 ≥ 9000, i=1, 2, 3 (amount to wholesaler 3)∑ x i4 ≥ 4500, i=1, 2, 3 (amount to wholesaler 4)∑ x i5 ≥ 3600, i=1, 2, 3 (amount to wholesaler 5)∑ x 1j ≤ 4500, j=1, 2, 3, 4, 5 (Production limit of plant 1)∑ x 2j ≤ 9000, j=1, 2, 3, 4, 5 (Production limit of plant 2)∑ x 3j ≤ 11250, j=1, 2, 3, 4, 5 (Production limit of plant 3)x 1j ≥ 02.8.5 Example 5Systems Analysis 41 Dr. Emad Elbeltagi